lecture23(complete)

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MA1100 Lecture 23 Cardinality Equivalence of sets Infinite sets Denumerable sets Countable sets Uncountable sets 1 Lecture 23 Chartrand: 10.1, 10.2, 10.3
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Lecture 23 2 Lesson Plan (countdown: 18 days) Tue Lecture Fri Lecture Tutorial Week 12 22. Number Th. 23. Cardinality Tut 9 Number Th. Week 13 24. Revision lecture HW5 25. Revision lecture Tut 10 Number Th./ Cardinality Week 14 READING WEEK Week 15 FINAL EXAM (NOV 24)
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Lecture 22 3 How far apart? Question For any positive integer n, we can find n consecutive composite numbers . Furthest : ? How far apart can two consecutive prime numbers be? Nearest : next to each other (2 & 3) As far apart as it can be. Proposition Proof: Tutorial 9 (Q11.9) This implies there are two consecutive primes that are at least n numbers apart
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Which set is larger? Question: They are both infinite sets Can we compare their “size”? N (0,1) 4 Lecture 23 the set of natural numbers the open interval from 0 to 1
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Equivalence of sets Proposition Let A and B be two sets. Define the relation: A º B if and only if there exists a bijection f: A Ø B from A to B. This is an equivalence relation. p q r a b c A B f x y z C g Sketch of Proof 5 Lecture 23 g o f f -1 reflexive A º A symmetric A º B B º A transitive A º B, B º C A º C a b c A I A
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Equivalent sets and cardinality Proposition Then A is equivalent to B if and only if |A| = |B|. Let A and B be finite sets. Construct a bijection f: A Ø B ( )If |A| = |B|, then A is equivalent to B . ( A is equivalent to B, then |A| = |B| . 6 Lecture 23 (See Lecture 18) If there exists a bijection f: A Ø B from A to B, we say A is (numerically) equivalent to B. Definition a 1 a 2 : a n b 1 b 2 : b n Define f(a i ) = b i for i = 1, 2, …, n
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Equivalent sets and cardinality Definition We define |A| = |B| if A is equivalent to B. Let A and B be infinite sets. there exists a bijection f: A Ø B So |A| |B| A is not equivalent to B. there is no bijection f: A Ø B For any function f: A Ø B, f is not a bijection 7 Lecture 23 A is equivalent to B |A| = |B|
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Positive odd numbers Example Let f: N Ø D + be defined by f(x) = 2x – 1 .
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture23(completeï&frac...

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